We compare two widely used regularizations.
The first is the logarithmic barrier function \(H(t) = -{\rm log}(t)\) ,
where (REF ) is the standard formulation of an
interior point method for (REF ).
This function is self-concordant, which is a key property in convex optimization.
The rank condition in (REF ) translates into polynomials
by taking numerators of all maximal minors. These define an
algebraic curve \(\mathcal {C}^{R,+}_{A,b,c}\)  in the polytope \(P_{A,b}\) . This is known as the
central path. Its starting point
\(x^*(\infty )\)  is the analytic center of \(P_{A,b}\) .
The algebraic complexity of these objects are governed by
the bounded regions in certain hyperplane arrangements.
See [1]}, [2]}.
